Toward computational oncology

: nonlinear simulation of centimeter-scale tumour growth in complex, heterogeneous tissues

Abstract

In this dissertation, we present three increasingly sophisticated mathematical models of solid tumor growth and new numerical techniques for accurately and e?ciently solving these models. In the first model, we simulate necrotic tumor growth into perfectly-vascularized, homogeneous tissue. We solve the model using a new level set/ghost fluid method that can produce accurate solutions on arbitrary domains, even when faced with challenging topological changes. This model provides a core framework for the development of more sophisticated models. After a brief presentation of a new geometry-aware curvature discretization for level set methods, we focus on a second model where we now include nutrient perfusion and proliferative pressure dissipation in the tissue surrounding the tumor. Using this model, we conduct a thorough study of the impact of the tumor microenvironment on tumor growth.We find that three characteristic morphologies emerge that depend primarily upon the microenvironment: invasive, fragmenting growth into nutrient-poor tissue; invasive, fingeringgrowth into nutrient-rich, biomechanically unresponsive tissue; and compact/hollow growthinto nutrient-rich, biomechanically responsive tissue. We discuss the implications of this finding on anti-angiogenic and anti-invasive cancer therapies. The third model treats tumor growth in complex, heterogeneous tissues using a non-linear nutrient equation and a two-sided pressure equation with geometric jump boundary conditions. We solve the model using a new level set/ghost cell method that can accurately and efficiently solve nonlinear elliptic PDEs on large, complex domains, even withgeometry-dependent jump boundary conditions. After testing the new technique, we simulate the growth of glioblastoma (an aggressive brain tumor) in a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white and gray matter, cerebrospinal fluid, and bone). We observe growth morphologies that are highly dependent upon the variable tissue characteristics - {an effect observed in real tumor growth. We close with a discussion of ongoing research, possible future extensions, the potential implications of our work, and the long-term goals of computational oncology. We outline some of the key mathematical, scientific, computational, and clinico-medical challenges that must be overcome before computational oncology can be accepted as a clinical tool for patient-tailored cancer therapy.

Date of Award	2007
Original language	English
Awarding Institution	University of California at Irvine
Sponsors	U.S Department of Education: Graduate Assistance in Areas of National Need , National Science Foundation & University of California at Irvine
Supervisor	John S. Lowengrub (Supervisor)